The production of hydrocarbons from subsurface formations typically commences by forming a borehole through the earth to a subsurface reservoir thought to contain hydrocarbons. From the borehole, various physical, chemical, and mechanical properties are "logged" for the purpose of determining the nature and characteristics, including for example, the porosity, permeability, saturation, and depth, of the subsurface formations encountered. One such logging technique commonly used in the industry is referred to as induction logging. Induction logging measures the conductivity or its inverse, the resistivity, of a formation. Formation conductivity is one possible indicator of the presence or absence of a significant accumulation of hydrocarbons because, generally speaking, hydrocarbons are relatively poor conductors of electricity. Formation water, on the other hand, typically salty, is a relatively good conductor of electricity. Thus, induction logging tools can obtain information that, properly interpreted, indicates the presence or absence of hydrocarbons.
These induction (also known as electromagnetic induction) well logging instruments were first introduced by Doll, H. G., "Introduction to Induction Logging and Application to Logging of Wells Drilled with Oil Based Mud," Journal of Petroleum Technology, June, 1949, pp. 148-62. Induction well logging instruments typically include a sonde having one or more transmitter coils and one or more receiver coils at axially spaced apart locations. Induction well logging instruments also typically include a source of alternating current (AC) which is conducted through the transmitter coils. The AC passing through the transmitter coils induces a magnetic field within the surrounding formation, causing the flow of eddy currents within the earth formations. In general, the magnitude of the eddy currents is proportional to the electrical conductivity (the inverse of the electrical resistivity) of the earth formations surrounding the instrument. The eddy currents, in turn, induce a magnetic field that is coupled to the receiver coil, thereby inducing in the receiver coil a voltage signal with magnitude and phase dependent upon the electrical characteristics of the adjacent formation.
Typically, the signal from the receiver coil is applied to one or more phase detection circuits, each of which generates a signal proportional to the magnitude of that component of the receiver coil signal having a particular, predetermined phase. Thus, one such phase detector circuit senses the magnitude of the component of the receiver coil signal that is in-phase with the transmitter current in the transmitter coil. This component signal is commonly referred to as the real or in-phase (R) component. A second phase detection circuit commonly used in induction logging tools detects the component of the receiver coil signal that is 90 degrees out of phase with the transmitter current. This latter component signal is commonly referred to as the quadrature-phase (X) component signal.
Because the output signal from the receiver coil is not itself an absolute measure of conductivity, but rather is merely proportional to the true formation conductivity, the output signal must be processed to obtain a log or plot of the true formation conductivity as a function of axial depth in the borehole. Most modern theoretical analysis of induction log processing is based upon the work of H. G. Doll which is summarized in his 1949 article. According to Doll's analysis, the in-phase component of the signal induced in the receiver coil is directly proportional to the conductivity of the surrounding formation, and the constant of proportionality, referred to by Doll as the "geometrical factor," is a function of the geometry of the tool as it relates to the portion of the formation being measured.
Doll calculated what he termed the "unit geometrical factor," which defines the relationship between the conductivity of a so-called "unit ground loop," a horizontal loop of homogeneous formational material having a circular shape with its center on the axis of the borehole and having a very small, square cross section, and the elementary voltage signal contributed by the unit ground loop to the total in-phase signal induced in the receiver coil. By integrating the unit geometrical factor across all unit ground loops lying within a horizontal plane spaced at some axial distance z from the center of the coil system, Doll obtained the geometrical factor for a "unit bed." A plot of this geometrical factor as a function of the axial distance from the center of the coil system gives what is commonly referred to as the "vertical geometrical factor` for the tool. It is an accurate plot of the sonde response function relating formation conductivity to output voltage measurements for the tool, assuming no attenuation or phase shift of the induced magnetic field as a consequence of the conductivity of the surrounding formation.
Induction logging technology has evolved significantly since its introduction by Doll. In recent years, induction devices consisting of several complex coil combinations have been replaced by tools with multiple arrays (see, for example, Beard, D. R., et al., "A New, Fully Digital, Full-spectrum Induction Device for Determining Accurate Resistivity with Enhanced Diagnostics and Data Integrity Verification," SPWLA 37.sup.th Annual Logging Symposium, June, 1996, Paper B; Beard, D. R., et al., "Practical Applications of a New Multichannel and Fully Digital Spectrum Induction System," SPE Annual Technical Conference and Exhibition, 1996, Paper No. 36504; and Barber, T. D., et al., "A Multiarray Induction Tool Optimized for Efficient Wellsite Operation," SPE 70.sup.th Annual Technical Conference and Exhibition, 1995, Paper No. 30583). Each array consists of one transmitter and a pair of receiver coils. These new induction devices are commonly referred to as array-type induction tools.
A simple induction array (two-coil array and three-coil array) responds to all its surrounding media, including formations, the borehole, and invasion zones if there are any. This response will be degraded by severe borehole effect and will suffer from low vertical and radial resolution. In order to avoid the weaknesses of the simple induction arrays, array combinations are used to increase the response contribution from the medium of interest, such as uninvaded formation, and at the same time to reduce the response contribution from the medium of disinterest, such as the borehole. This process by which the output of an induction logging instrument is made to effectively zoom in on a specific space of its surrounding medium and mute the unwanted peripherals is referred to as focusing.
The older style tools attempt to focus the tool response using carefully selected coil arrangements. The focusing therefore is fixed by the tool design, i.e. these tools are "hardware-focused". In array-type induction tools, the measurements from various arrays are combined through a software algorithm to achieve focusing of the signal response, i.e. these tools are "software-focused". This processing produces a set of curves with predetermined depth of investigation, vertical resolution and other optimized 2D features.
Using software-based focusing provides greater flexibility for handling various logging environments and for creating more reliable induction logs. However, the quality and accuracy of the final focused logs are dependant on the accuracy of the software focusing method.
The current focusing method was proposed by Barber and Zhou (see Barber, T. D. and Rosthal, R. A., "Using a Multiarray Induction Tool to Achieve High-Resolution Logs with Minimum Environmental Effects," SPE 66.sup.th Annual Technical Conference and Exhibition, 1991, Paper No. 22725 and Zhou, Q., Beard, D. and Tabrovsky L., "Numerical Focusing of Induction Logging Measurements," 12.sup.th Workshop in Electromagnetic Induction in Earth, August, 1994) and is, for reference purposes, here referred to as the "conventional focusing method". The conventional focusing method can be expressed mathematically as ##EQU1##
where .sigma..sub.ai is the measured log from the i.sup.th array; W.sub.i is the focusing filter; m.sub.ary is the total number of arrays; and Z.sub.min and Z.sub.max define the depth window surrounding the output point.
Theoretically, the software focusing method described by equation (1) can be traced back to the Born Approximation (a linear approximation of the measured response of a medium) and then the condition for equation (1) is an assumption of an homogeneous background. Practically, the current focusing method (conventional focusing method) produces good quality focused logs when the formation conductivity varies with small to moderate contrasts between adjacent formation beds. However, when the formation conductivity varies with very large conductivity contrasts, i.e. if the formation is very "inhomogeneous", the focused logs are not as good as would be expected.